L(s) = 1 | + 1.06·2-s + 1.32·3-s − 0.861·4-s + 5-s + 1.41·6-s + 3.66·7-s − 3.05·8-s − 1.24·9-s + 1.06·10-s + 1.40·11-s − 1.14·12-s + 2.40·13-s + 3.91·14-s + 1.32·15-s − 1.53·16-s + 6.20·17-s − 1.32·18-s − 2.86·19-s − 0.861·20-s + 4.85·21-s + 1.49·22-s + 3.58·23-s − 4.04·24-s + 25-s + 2.56·26-s − 5.62·27-s − 3.15·28-s + ⋯ |
L(s) = 1 | + 0.754·2-s + 0.764·3-s − 0.430·4-s + 0.447·5-s + 0.577·6-s + 1.38·7-s − 1.07·8-s − 0.415·9-s + 0.337·10-s + 0.422·11-s − 0.329·12-s + 0.667·13-s + 1.04·14-s + 0.342·15-s − 0.383·16-s + 1.50·17-s − 0.313·18-s − 0.656·19-s − 0.192·20-s + 1.05·21-s + 0.319·22-s + 0.746·23-s − 0.825·24-s + 0.200·25-s + 0.504·26-s − 1.08·27-s − 0.596·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.618373389\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.618373389\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 3 | \( 1 - 1.32T + 3T^{2} \) |
| 7 | \( 1 - 3.66T + 7T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 13 | \( 1 - 2.40T + 13T^{2} \) |
| 17 | \( 1 - 6.20T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 3.58T + 23T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 - 2.44T + 31T^{2} \) |
| 37 | \( 1 + 6.49T + 37T^{2} \) |
| 41 | \( 1 - 7.04T + 41T^{2} \) |
| 43 | \( 1 - 8.96T + 43T^{2} \) |
| 47 | \( 1 + 7.50T + 47T^{2} \) |
| 53 | \( 1 + 7.59T + 53T^{2} \) |
| 59 | \( 1 - 3.25T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 7.69T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 6.29T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 4.00T + 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.108541277278157176349135774747, −8.278109367880977488780852616092, −7.960963427802265469517048887725, −6.65407216265158907996007748329, −5.60964620722318973644331305535, −5.22328201281635663065218577476, −4.14442137006100140946833341183, −3.45020354475282489843123487863, −2.42857677778047576984823084983, −1.23026845697656184821513572852,
1.23026845697656184821513572852, 2.42857677778047576984823084983, 3.45020354475282489843123487863, 4.14442137006100140946833341183, 5.22328201281635663065218577476, 5.60964620722318973644331305535, 6.65407216265158907996007748329, 7.960963427802265469517048887725, 8.278109367880977488780852616092, 9.108541277278157176349135774747